**JPK60 | Random Matrices from Quantum Chaos to the Riemann Zeta Function:**

**A Celebration in Honour of Jon Keating’s 60th Birthday**

**5 – 7 July 2023**

School of Mathematics, University of Bristol, UK

Random Matrix Theory (RMT) is ubiquitous in the mathematical and physical sciences because of its broad range of applications and its predictive power, which allow accurate calculations and asymptotic analysis that are not accessible through traditional techniques. The interdisciplinary nature of RMT is epitomized by Jon Keating’s career, which has been characterized by the rare ability of initiating new areas of research by bringing together areas of mathematics that on the surface have little in common. This conference will feature leading mathematicians working at the interface of quantum chaos, analytic number theory, probability and random matrix theory.

**Organisers: **

Emma Bailey (CUNY)

Tamara Grava (Bristol)

Francesco Mezzadri (Bristol)

Nina Snaith (Bristol)

Brian Winn (Loughborough)

Public Lecture by Persi Diaconis (Stanford) on** **The Mathematics of Solitaire

**Speakers:**

**Louis-Pierre Arguin **(CUNY): **The Fyodorov-Hiary-Keating Conjectures**

In 2012, Fyodorov, Hiary & Keating and Fyodorov & Keating proposed a series of conjectures describing the statistics of large values of zeta in short intervals of the critical line. In particular, they relate these statistics to the ones of log-correlated Gaussian fields and of characteristic polynomials of random matrices. In this talk, I will present recent results that answer many aspects of these conjectures, including upper and lower bounds for the maximum that are sharp up to the order of fluctuations. The connections with log-correlated Gaussian fields, in particular with the work of Bramson on branching Brownian motion, will be emphasized. Based on joint works with P. Bourgade and M. Radziwill.

**Emma Bailey** (CUNY): **On the Large Deviations of Selberg’s Central Limit Theorem**

Selberg’s central limit theorem proves that the logarithm of the Riemann zeta function evaluated at a uniform random height (drawn between T and 2T) on the critical line behaves as a complex Gaussian random variable with variance log(log T). In prior work joint with Arguin, we established an upper bound on the right-tail large deviations of the real part of this complex random variable, at scales on the order of the standard deviation. In this talk we will present recent work that proves a matching lower bound for the real part of the logarithm of the Riemann zeta function, evaluated at a random point at a horizontal distance alpha off the critical line, where alpha is a suitable multiple of 1/log T. Additionally we will discuss results, both theoretical and numerical, on the joint large deviations of the real and imaginary parts of the original complex random variable. This talk features work joint with L.-P. Arguin and D. Blyschak.

**Michael Berry** (Bristol): **Four Geometrical-Optics Illusions**

Centuries after the laws of geometrical optics were established, they still have nontrivial and varied applications. Illustrating this are some illusions:

*Mirages*, and Raman’s error. Understanding why he denied the applicability of geometrical optics requires careful exploration of the continuum limit of a discretely-stratified medium, to reveal its nonuniform convergence.

*Oriental magic mirrors and the Laplacian image*. The optics of these several-millennia-old objects involves the unfamiliar regime of pre-focal brightening. The transmission analogue (‘Magic windows’) raises a challenge for freeform optics.

*The squint moon and the witch ball.* The moon sometimes appears to point the wrong way because we perceive the sphere of directions as a distorted ‘skyview’, on which geodesics appear curved. This can be conveniently viewed and analysed by viewing the sky in a reflecting sphere.

*Distorted and topologically disrupted reflections in curved mirrors*. Mirror-reflected rays from each point of a continuous object form caustic surfaces in the air. Images are organised by those points whose caustics intersect our eyes, and can be systematically understood in terms of the elementary catastrophes of singularity theory.

**Brian Conrey **(AIM): **Averages of Characteristic Polynomials in Random Matrix Theory**

After the seminal works of Keating and Snaith and of Katz and Sarnak we know that averages of products and ratios of products of shifted characteristic polynomials over the classical compact groups model similar averages of L-functions in families. Work of Bogomolny and Keating from the 1990s gave a heuristic approach to the GUE conjecture assuming the Hardy-Littlewood conjectures. In a series of papers Conrey and Keating adapted this work to show how to obtain the “recipe” for moments from divisor correlations. In this talk we will describe work with Baluyot on the analogue of the C-K papers but in Random Matrix Theory.

**Neil O’Connell **(UCD): **A Markov Chain on Reverse Plane Partitions **

I will discuss a natural Markov chain on reverse plane partitions which is closely related to the Toda lattice.

**Persi Diaconis** (Stanford): **Public Lecture** on **The Mathematics of Solitaire**

One of the embarrassing facts about probability theory: we don’t know the odds of winning at solitaire! I mean ordinary klondike, played on computer screens and cellphones millions of times a day. For example, in Vegas, you can ‘buy a deck for $52 and get $5 for each card turned up on the ace piles. Is this anything like a fair game? I will review what we know (after all, it’s 2023 and the computer is here–surely they know how to play solitaire (nope)). I’ll turn to what we always turn to, Polya’s dictum ‘If there is a hard problem you can’t solve there is an easier problem you can’t solve.”” Patience sorting is a simpler form of solitaire and here mathematics can be brought in. The mathematics is hard and interesting (and gives definitive answers involving one main theme of our conference–random matrix theory). Even here, bending the rules back towards Klondike leads to easy to understand and wide open problems.”

**Alexandra Florea **(UC Irvine): **Negative Moments of the Riemann Zeta-Function**

I will talk about joint work with H. Bui regarding negative moments of the Riemann zeta-function. I will explain how to obtain asymptotic formulas in certain ranges for the shift in the zeta-function in the denominator, and will discuss some applications to the question of obtaining cancellation in partial sums of the generalized Mobius function.

**Yan Fyodorov** (KCL): **Replica-Symmetry Breaking Transitions in the Large Deviations of the Ground-State of a Spherical Spin-Glass**

The ground state (minimal energy) of the simplest p=2 spherical spin glass coincides with the largest eigenvalue of the GOE matrix, hence the associated Large deviations (and, on a different scale, Tracy-Widom distribution) provide a natural starting point for studying fluctuations of the minimal energy in a general spherical spin glass. We use replica method to develop a theory of Large Deviations of rate N for the spherical spin glass model with random magnetic field and reveal a surprisingly rich picture of Replica Symmetry Breaking (RSB) phenomena required to describe such Large Deviations. Our main qualitative conclusion is that the level of RSB governing the Large Deviations may be different from that for the typical ground state. The talk will be based on a joint work with Bertrand Lacroix-A-Chez-Toine and Pierre Le Doussal.

**Alice Guionnet **(Lyon): **Large Deviations for Large Random Matrices**

I will discuss recent developments in the theory of large deviations in random matrix theory and their applications.

**Alexander Its** (IUPUI): **Toeplitz and Hankel Determinants, and Random Matrices. A Riemann-Hilbert Point of View**

We review some of the old and new results concerning the asymptotic analysis of Toeplitz and Hankel determinants and their use in the theory of random matrices. The main focus will be in the ”Riemann-Hilbert” side of the story. Special attention will be put to the application emerged from the work Jon Keating.

**Jens Marklof **(Bristol): **Statistics of Directions**

Given an infinite point set in Rn, say, it is natural to investigate the distribution of directions in which points appear to a fixed observer. Perhaps surprisingly, even strongly correlated sets such as lattices and quasicrystals show intriguing limit distributions: these follow -in some instances- from an application of Ratner’s celebrated measure classification theorem for unipotent flows. I will survey some of the key results on directions in both Euclidean and hyperbolic geometry, and explain their connection to fundamental questions in dynamical systems and number theory.

**Zeev Rudnick **(Tel Aviv): **The Robin Eigenvalue Problem: Statistics and Arithmetic**

Robin boundary conditions are used in heat conductance theory to interpolate between a perfectly insulating boundary, described by Neumann boundary conditions, and a temperature fixing boundary, described by Dirichlet boundary conditions. I shall explore the statistics of these Robin eigenvalues for planar domains, and the fluctuations of the gaps between the Robin and Neumann spectrum, in part driven by numerical experimentation. I will display connections with questions from number theory and from “”quantum chaos”” and discuss some tantalizing open problems. Based on joint work with Igor Wigman and Nadav Yesha.”

**Peter Sarnak **(Princeton): **Root Numbers and Murmurations**

Online presentation (no recording)

We discuss statistical correlations coined “murmurations” uncovered by machine learning routines run on the database of elliptic curves .

**Nick Simm **(Sussex): **Character Expansion in Non-Hermitian Ensembles **

The archetypal model of a non-Hermitian random matrix is the Ginibre ensemble, consisting of i.i.d. standard Gaussian entries with no symmetry constraints. I will discuss the character expansion technique for evaluating correlations of characteristic polynomials in such models. By employing the theory of symmetric functions, particularly Schur polynomials, we give a unified treatment in the real, complex and quaternion settings. In the case of the real Ginibre ensemble, this yields an explicit Pfaffian expression which is well suited to studying asymptotics.

*This is joint work with A. Serebryakov (Sussex).*

**Kannan Soundararajan** (Stanford): **Central Limit Theorems for Random Multiplicative Functions**

I will discuss some problems related to understanding the distribution of sums of random multiplicative functions. The motivation for considering such functions is to gain insight on the behavior of deterministic number theoretic functions such as Dirichlet characters or the Mobius and Liouville functions. Some subtleties in the distribution of these random functions have emerged that are closely related to the work of Fyodorov—Hiary—Keating and Bailey—Keating on the distribution of the zeta-function in randomly chosen short intervals.

This conference received financial support from the American Institute of Mathematics (AIM), the Heilbronn Institute for Mathematical Research (HIMR), and the London Mathematical Society (LMS).

The conference was also supported by the National Science Foundation (NSF) (award number 2306332). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.